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q-invariance at the crystal limit: isomorphism between C0(SU(n+1)) and Cq(SU(n+1)) for 0<q<1

Determine whether the crystallized C*-algebra C0(SU(n+1)) is isomorphic to Cq(SU(n+1)) for all q in (0,1), thereby establishing q-invariance across the crystal limit for SU(n+1).

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Background

For SU(n+1) at nonzero q, Giselsson [7] proved that the C*-algebras Cq(SU(n+1)) are mutually isomorphic for all q in (0,1), establishing q-independence in that range. The natural next question is whether this q-independence persists through the crystal limit q→0.

The authors highlight this as unresolved and note that their realization of the crystallized algebra as operators on the same space as the Soibelman representation may help to settle the question.

References

So the question that remains to be settled in this case is whether the crystallized algebra CpSU p0`1q is isomorphic to the ones for q P p0,1q.

— Irreducible representations of the crystallization of the quantized function algebras $C(SU_{q}(n+1))$ (2402.09347 - Giri et al., 14 Feb 2024) in Introduction